$\lambda$-shaped random matrices, $\lambda$-plane trees, and $\lambda$-Dyck paths

We consider random matrices whose shape is the dilation $N\lambda$ of a self-conjugate Young diagram $\lambda$. In the large-$N$ limit, the empirical distribution of the squared singular values converges almost surely to a probability distribution $F^{\lambda }$. The moments of $F^{\lambda }$ enumerate two combinatorial objects: $\lambda$-plane trees and $\lambda$-Dyck paths, which we introduce and show to be in bijection. We also prove that the distribution $F^{\lambda }$ is algebraic, in the sense of Rao and Edelman. In the case of fat hook shapes we provide explicit formulae for $F^{\lambda }$ and we express it as a free convolution of two measures involving a Marchenko-Pastur and a Bernoulli distribution.